The way things are handled now (as far as I’m aware, not a judge but still a Rules Advisor), if upkeep happens and Huntmaster’s flip conditions were met, it’s supposed to flip. If your opponent has drawn their normal card for the turn (and thus signified the end of the upkeep phase), then they’ve missed the trigger. You aren’t obligated to remind them and play will continue as if the flip conditions weren’t met.

If they then remember during first main phase, combat, whenever, then they have to ask you if you (as their opponent) if it flips. You may choose to let it flip or not depending on whichever you feel is more beneficial for you.

Given Huntmaster’s flip isn’t strictly negative (there are benefits every time it flips), your opponent won’t get a warning.